Our proof of Theorem REF  differs only slightly from the one given in [1]}, providing a minor improvement of Theorem REF .
In fact, Theorem REF  suffices to conclude that, for \(K\) -varieties \(X\)  and \(Y\)  satisfying the Hilbert property over \(K\) , the product \(X\times _K Y\)  also has the Hilbert property over \(K\) , but not for the analogue over an integral domain \(R\)  with fraction field \(K\) .
The problem in the situation of Theorem REF  is that the Hilbert property of a fibre
\(X_{K,s}\)  over \(s\in S(R)^{(1)}\)  does not give us near-\(R\) -integral points of \(X\) , and asking \(X_s\)  to satisfy the Hilbert property over \(R\)  is too
strong, because instead of points in \(X_s(R)^{(1)}\) , we are interested in points in \(X(R)^{(1)}\)  contained in \(X_{K,s}\) . This leads to the following natural generalization
of Theorem REF .
